Problem 34 - Fundamentals of Mathematics

Question

For the following problems, find each part without using a diagram. $$\frac{1}{2}\( of \)\frac{4}{5}$$

Step-by-Step Solution

Answer

The product is \( \frac{2}{5} \).

Step 1: Break Down the Problem

We need to find the product of \( \frac{1}{2} \) and \( \frac{4}{5} \). This means we multiply these two fractions together. The problem involves multiplying two fractions.

Step 2: Multiply the Numerators

To multiply fractions, start by multiplying the numerators (the top numbers) together. So multiply 1 by 4: \[ 1 \times 4 = 4 \]

Step 3: Multiply the Denominators

Next, multiply the denominators (the bottom numbers) together. Multiply 2 by 5: \[ 2 \times 5 = 10 \]

Step 4: Form the Product Fraction

Combine the results from steps 2 and 3 to form the new fraction. Using the products from the previous steps, we have: \[ \frac{4}{10} \]

Step 5: Simplify the Fraction

To simplify \( \frac{4}{10} \), find the greatest common divisor (GCD) of 4 and 10, which is 2. Divide the numerator and the denominator by this number: \[ \frac{4 \div 2}{10 \div 2} = \frac{2}{5} \]

Key Concepts

Understanding the Numerator

In any fraction, the numerator is the number placed above the fraction line. It indicates how many parts we are considering out of the whole. For example, in our problem, the fraction \( \frac{1}{2} \) has a numerator of 1. This means we have one part of the whole divided into two parts. Similarly, \( \frac{4}{5} \) has 4 as its numerator, indicating four parts out of five.

When multiplying fractions, we first multiply the numerators of both fractions together to find the numerator of the product. So, for \( \frac{1}{2} \times \frac{4}{5} \), the calculation is simple: \( 1 \times 4 = 4 \). This product becomes the numerator of the new fraction.

The Importance of the Denominator

The denominator is the number located below the fraction line. It shows into how many total parts the whole is divided. In our case, \( \frac{1}{2} \) has a denominator of 2, which tells us the whole is divided into two parts. For \( \frac{4}{5} \), the denominator is 5, meaning the whole is split into five parts.

When multiplying fractions, we multiply the denominators to find the new denominator. Applying this to our fractions gives \( 2 \times 5 = 10 \). So, the denominator of our resulting fraction after multiplication is 10, resulting in \( \frac{4}{10} \).

• This keeps the fraction standardized, making it easier for further calculations.

Steps for Simplifying Fractions

Fractions can often be reduced to a simpler form. This means expressing the fraction with smaller, easier-to-manage numbers without changing its value. We follow these steps to simplify:

• Identify the numerator and denominator. For instance, in \( \frac{4}{10} \).
• Determine the greatest common divisor (GCD) of these numbers.
• Divide both the numerator and the denominator by their GCD.

For our example, the fraction \( \frac{4}{10} \) can be simplified by dividing both 4 and 10 by their GCD, which is 2. This results in \( \frac{2}{5} \), a much cleaner fraction to work with.

Finding the Greatest Common Divisor (GCD)

The greatest common divisor is the highest number that divides exactly into two or more numbers. Finding the GCD is crucial for simplifying fractions. Here's how you do it:

To find the GCD of two numbers, identify all the factors of each number. Compare these lists and choose the largest number that appears in both lists.

• For 4: the factors are 1, 2, and 4.
• For 10: the factors are 1, 2, 5, and 10.

The largest common factor between them is 2, so the GCD of 4 and 10 is 2.

Using this GCD to divide the numerator and the denominator gives us the simple form \( \frac{2}{5} \). Finding the GCD helps in reducing fractions to their simplest form efficiently.